Horizontal And Vertical Graph Stretches and Compressions Part 2 of 3

Okay, in this video,
I wanna talk more about vertical and

horizontal stretching and reflecting.

And I'm gonna definitely talk
about case 3 and case 4,

which deals with horizontal stretches or
compressions.

And then 5 or 6 deals with reflections,
maybe we'll get to that one, we'll see.

Again, the basic idea and, 3 with these
horizontal compressions or stretches.

If you multiply by a number
bigger than 1 on the inside,

it's actually gonna squish
your graph together.

If you multiply by a number between 0 and
1,

it's gonna actually pull it apart, okay?

So let's look at at least two examples.

So again, here's my original graph,
here at the top,

my blue graph, this is the one I'm
gonna tweak to come up with a new one.

So it's this little sawtooth function,
-4, 0, -3, -2, -2,

0, -1, 2, 0, 0, and
then it's this little step function, okay?

So, what I'm gonna graph now
is the function f of 2x, and

what this again does,
since I'm multiplying the xs by 2,

it's actually gonna compress the graph.

It's gonna basically compress
it by a factor of 2.

So instead of going out from 0 to -4 and
0 to 4,

it's now gonna go from 0 to -2 and 0 to 2.

I think this is the one that always
confuses people as well, cuz, hey,

you're multiplying by 2,
that should make things bigger.

But you can't really think about it
like that, or I guess if you do so,

it's not correct.

Think about it this way, if I plug -1 in,

if I let x equal -1,
what would I get on the inside?

I would get a value of -2, but
according to the original graph,

it says if you plug -2 in,
you should get 0 out.

So if I plug -1 in,
I'm gonna get -2 on the inside,

which should give me an output of 0,
okay, and that's the basic idea.

You're gonna kind of cut
the x coordinates in half.

So, originally,
at -2 there was an x coordinate of 0.

If I cut that x coordinate in half It now
becomes -1, keep the same y coordinate.

At -4, the original x coordinate is -4,

if you cut that in half, you'll get -2.

Okay, originally,
at- 1 it was up here at a y value of 2.

If you cut -1 in half you're at negative
one-half, and then I'm up here at 2,

likewise, at negative three-halves,
I'm gonna be down here at -2, okay?

So it still has the same height,
but everything has

gotten squished together by a factor of 2,
okay.

And then that's gonna be the same thing
that happens on the right hand side.

Instead of extending out
a distance of 0 to 2,

it's only gonna go out from 0 to 1.

And then instead of going out from 2 to 4,
it's only gonna go out from now 1 to 2.

Okay, again, imagine chopping the interval
0 to two-half, you get 0 to 1.

Imagine chopping
the interval 2 to 4 in half,

you would get the interval 1 to 2, okay?

So, again,
notice the heights are the same, but

it should look definitely
a little more squished together.

Okay, so let's do another one of these.

Now, what I'm gonna do is, again,
very much a similar thing.

I'm gonna multiply now
the inside by one-half, and now,

if you multiply by a one half, instead
of compressing it by a factor of 2,

you actually stretch it by a factor of 2.

So I can't kind of keep
the scale correct on this graph,

cuz I've got -4 to 4.

So now the idea is,
instead of getting squished together,

you're gonna pull out by a factor of 2.

So that means, I'm gonna go,
instead of from out to -4,

I'm gonna go all the way out here to -8,

and instead of +4,
I'm gonna go all the way out here to +8.

Okay, and now basically,
you do the same thing, originally, at -2,

that's where I got a 0.

If you multiply that x coordinate by 2,
cuz we're stretching

it by a factor of 2,
you're gonna go out to -4, okay?

And at-1, you originally had
a y coordinate of 2, well,

now if you multiply that x coordinate
by 2, you're gonna be at -2.

And then we'll be up here
at a height of 2, likewise,

at -6, we're gonna be down
here at a height of -2.

Again, if you multiply that
original x coordinate by 2,

you're gonna keep that same y value, but

the x value turns into -6, and
the x value stays the same.

So again, play connect the dots.

Obviously, again, I'm not a great artist,
so forgive my artistry.

Again, it doesn't really
look stretched out because

just the proportions of my graph.

But if you had a bigger piece of paper,
I think you would definitely see

this thing looking more elongated,
so try that for yourself, and

then we'll go from, whoops,
we'll go all the way out to 4 this time.

So instead of being this little
horizontal line at -1 from 0 to 2,

again, now I double that, so it's gonna be
looking like that all the way out to 4.

And then I jump down to -2 at 4, and that
extends all the way out to the value of 8.

Okay, so this would be the graph,
again, of f one half of x.

And I'm definitely gonna put all of
this stuff together in some more

concrete examples.

Again, just trying to give you a general
idea of what's going on, let's see.

I don't know if I can do
the other two real quick.

We'll save the other two for
one other video.

So this, again, deals with horizontal
stretches or compressions.

Next, I'll basically deal with
flips about the x-axis and

flips about the y-axis, so
look for another video.

And then again,

stay tuned I'll do some more general
ones where I do all the compressions,

and stretching ,and rotating, and
shifting, and all of that stuff combined.